Using nomenclature that is well known to those of ordinary skill in the art of industrial vibration analysis, the frequency response problem for damped structures discretized by the finite element method can be defined by Equation (1), having the general form:{−ω2M+iωB+[(1+iγ)K+i(K4)]}X=F.  (1)Here ω is the radian frequency of the time-harmonic excitation and response, and i=√(−1). M is the mass matrix, which may be symmetric. B is the viscous damping matrix, which may be nonsymmetric if gyroscopic effects are modeled. It is important to note that if the structures being modeled are automobiles, the matrix B may be of very low rank (e.g., less than about 50, including zero) because B's rank is substantially equal to the number of viscous damping elements, which can include shock absorbers and engine mounts. If gyroscopic effects are modeled, the rank of B is not ordinarily greatly increased.
The scalar γ is a global structural damping coefficient. K is the symmetric stiffness matrix. K4 is a structural damping matrix that may be symmetric and represents local departures from the global structural damping level represented by γ.
X is a matrix of displacement vectors to be determined in the frequency response analysis. Each vector in X represents a response of the structure to the corresponding load vector in the matrix F. The matrices M, B, K and K4 may be real-valued and sparse, with millions of rows and columns. The response matrix X can be dense and complex-valued, although only a small number of rows in X, associated with specific degrees of freedom for the structure, may be of interest.
In automotive applications, the frequency response problem can be solved at hundreds of frequencies to obtain frequency response functions over a broad frequency range. Direct or iterative solutions at each frequency, using a different coefficient matrix, is not usually feasible. The practical approach, therefore, has been to approximate the solution using the subspace of undamped natural modes of vibration having natural frequencies lower than a specified cutoff frequency. Because the number of modes m is usually much smaller than the dimension of the original frequency response problem in Equation (1), the solution of the problem typically becomes more economical.
These modes are obtained as a partial eigensolution of the generalized eigenvalue problem KΦ=MΦΛ, in which Φ is a rectangular matrix whose columns may be eigenvectors, and Λ is a diagonal matrix containing real-valued eigenvalues, which may be squares of natural frequencies. With mass normalization, so that ΦTMΦ=I, where I is the identity matrix, Φ and Λ satisfy ΦTKΦ=Λ. By making the approximation X≈ΦY and pre-multiplying the frequency response problem of Equation (1) by ΦT, the modal frequency response problem, which is of dimension m, may be obtained in Equation (2) as:{−ω2I+iω(ΦTBΦ)+[(1+iγ)Λ+i(ΦTK4Φ)]}Y=ΦTF.  (2)The accuracy of this modal approximation may be adequate for certain purposes if the cutoff frequency is chosen appropriately.
The number of modes m represented in Φ can be in the thousands, which may reduce the dimension of the original frequency response problem of Equation (1) to that of the modal frequency response problem of Equation (2). When B and K4 are not present, the solution of the modal frequency response problem can be less costly since the coefficient matrix might be diagonal. However, when B and K4 are nonzero, the matrices ΦTBΦ and ΦTK4Φ may each be of dimension m×m and fully populated, such that solving the modal frequency response problem becomes considerably more expensive.
This is because, at every frequency, a complex dense square matrix having a dimension in the thousands may have to be factored. The cost of factorization may be proportional to the cube of the matrix dimension, which can be equal to the number of eigenvectors in Φ. As the upper frequency limit for the analysis increases, so does the cutoff frequency for the modes represented in Φ, so the number of modes m may increase markedly. Therefore, the expense of the modal frequency response analysis can increase rapidly as the upper frequency limit for the analysis increases. Until recently, frequency response analysis in industry has generally been limited to low frequencies, lessening this concern, mostly because of the cost of the KΦ=MΦΛ eigensolution.
Another approach to the solution of the modal frequency response problem is to diagonalize the coefficient matrix by solving an eigenvalue problem. Because the coefficient matrix may be quadratic in frequency, a state-space formulation might be used, in which the unknowns include velocities in addition to displacements. However, this may result in doubling the dimension of the eigenvalue problem, possibly increasing the cost of the eigensolution by a factor of eight. Complex arithmetic and asymmetry can also add to the cost.
Making use of such an eigensolution approach might be more economical in some cases than the approach of factoring the complex dense coefficient matrix in Equation (2) at each frequency, but the difference in cost has not been enough to motivate implementing the eigensolution approach in most industrial structural analysis software. Thus, there is a need for apparatus, systems, articles, and methods for more efficiently determining the frequency response characteristics of damped structures, including vehicles, such as automobiles, aircraft, ships, submarines, and spacecraft.